Reflection Resonances in the One-Dimensional Anderson Localization: Finite-Length Statistics, Wigner Time Delay, and Boundary Eigenfunctions

Abstract

We study reflection-resonance poles $Z_j=E_j-i\Gamma_j$, $\Gamma_j>0$, of a finite one-dimensional disordered sample of length $L$, coupled at one end to a semi-infinite lead, in the regime $L\gg\ell_L\gg k^{-1}$, where $\ell_L$ is the localization length and $k=\sqrt{E}$. The key step is to relate the resonance density to the reflection coefficient at the complex energy $E+i\Gamma$, corresponding to uniform absorption. Exact finite-chain Kac--Rice and Poincar\'e--Lelong identities reduce pole counting to a finite-length diffusion of the reflected intensity. For the perfect contact transparency the density crosses over from the localization-controlled $\Gamma^{-1}$ law to the broad-resonance $\Gamma^{-2}$ law at $\Gamma_L=k/\ell_L$, in agreement with $L\to\infty$ result of Fyodorov and Meibohm. Finite length cuts off the $\Gamma^{-1}$ regime at $\Gamma_{\rm ultra}=\frac{e^{1-\gamma_{\rm E}}}{2}\Gamma_L e^{-L/\ell_L}$. We derive the ultranarrow resonances crossover shape as an explicit moving front; for contact transparency $\mathcal T<1$ this scale shifts to $\mathcal T\Gamma_{\rm ultra}$. The same reflection process yields the finite-length Wigner time-delay statistics and, in the weak-absorption limit, the Comtet--Texier distribution. We show that at eigenvalues of the corresponding closed Dirichlet sample the Wigner delay is inversely proportional to the squared boundary derivative of the normalized eigenfunction. This quantity also gives the eigenvalue response to displacement of the Dirichlet boundary at the lead-contact end, and hence the force exerted by the eigenmode on that boundary; we obtain its finite-length distribution. Finally, the finite-transparency resonance density obeys the single-channel Moldauer--Simonius sum rule, linking its perfect-coupling divergence to the $\Gamma^{-2}$ tail. Direct lattice and spectral computations test the crossover and front constant.